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We establish the time-asymptotic stability of solutions to theone-dimensional compressible Navier-Stokes-Fourier equations, with initial dataperturbed from Riemann data that forms a generic Riemann solution. The Riemannsolution under consideration is composed of a viscous shock, a viscous contactwave, and a rarefaction wave. We prove that the perturbed solution ofNavier-Stokes-Fourier converges, uniformly in space as time goes to infinity,to a viscous ansatz composed of viscous shock with time-dependent shift, aviscous contact wave and an inviscid rarefaction wave.  This is a first resolution of the challenging open problem associated withthe generic Riemann solution. Our approach relies on the method ofa-contraction with shifts, specifically applied to both the shock wave and thecontact discontinuity wave. It enables the application of a global energymethod for the generic combination of three waves.

Time-asymptotic stability of generic Riemann solutions for compressible Navier-Stokes-Fourier equations